<p>We consider the Dirichlet problem for a class of second-order nonlinear degenerate elliptic equations with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-right-hand side in a bounded open set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varOmega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>). The coefficients of the equations satisfy the growth and coercivity conditions involving a growth parameter <i>p</i> and a weighted function <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu \in L^\infty (\varOmega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that the set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\{t>0:1/{\mu }\in L^t(\varOmega )\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>μ</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>t</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is nonempty. We prove a number of results on the existence and nonexistence of weak solutions of the given problem. The value <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$t_{\mu }=\sup \{t>0:1/{\mu }\in L^t(\varOmega )\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mi>μ</mi> </msub> <mo>=</mo> <mo movablelimits="true">sup</mo> <mrow> <mo stretchy="false">{</mo> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>μ</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>t</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> plays an important role in our study. In particular, we describe cases where the inequality <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$p>2-1/n+1/t_{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msub> <mi>t</mi> <mi>μ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is necessary and sufficient for the existence of a weak solution of the considered problem for every right-hand side in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^1(\varOmega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the existence and nonexistence of weak solutions for a class of degenerate elliptic equations with $L^1$-data

期刊论文
发表日期：

2026年1月19日

查看全文

Alexander A. Kovalevsky

摘要

We consider the Dirichlet problem for a class of second-order nonlinear degenerate elliptic equations with $L^1$ $L^{1}$ -right-hand side in a bounded open set $\varOmega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ ( $n\geqslant 2$ $n ⩾ 2$ ). The coefficients of the equations satisfy the growth and coercivity conditions involving a growth parameter p and a weighted function $\mu \in L^\infty (\varOmega )$ $μ \in L^{\infty} (Ω)$ such that the set $\{t>0:1/{\mu }\in L^t(\varOmega )\}$ ${t > 0 : 1 / μ \in L^{t} (Ω)}$ is nonempty. We prove a number of results on the existence and nonexistence of weak solutions of the given problem. The value $t_{\mu }=\sup \{t>0:1/{\mu }\in L^t(\varOmega )\}$ $t_{μ} = sup {t > 0 : 1 / μ \in L^{t} (Ω)}$ plays an important role in our study. In particular, we describe cases where the inequality $p>2-1/n+1/t_{\mu }$ $p > 2 - 1 / n + 1 / t_{μ}$ is necessary and sufficient for the existence of a weak solution of the considered problem for every right-hand side in $L^1(\varOmega )$ $L^{1} (Ω)$ .

On the existence and nonexistence of weak solutions for a class of degenerate elliptic equations with \(L^1\)-data

摘要